Cuspidal integrals for SL(3)∕Kϵ

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Cuspidal integrals for SL(3)/K ϵ Mogens Flensted-Jensen a , Job J. Kuit b , ∗ a Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø,

Denmark b Institut für Mathematik, Universität Paderborn, Warburger Straße 100, 33089 Paderborn, Germany

Received 8 February 2018; received in revised form 11 May 2018; accepted 16 May 2018 Communicated by E.P. van den Ban

Abstract We show that for the symmetric spaces SL(3, R)/SO(1, 2)e and SL(3, C)/SU(1, 2) the cuspidal integrals are absolutely convergent. We further determine the behavior of the corresponding Radon transforms and relate the kernels of the Radon transforms to the different series of representations occurring in the Plancherel decomposition of these spaces. Finally we show that for the symmetric space SL(3, H)/Sp(1, 2) the cuspidal integrals are not convergent for all Schwartz functions. c 2018 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝

0. Introduction In this article we investigate the notion of cusp forms for some symmetric spaces of split rank 2. Harish-Chandra defined a notion of cusp forms for reductive Lie groups and he showed that the space of cusp forms coincides with the closure in the Schwartz space of the span of the discrete series of representations. This fact plays an important role in his work on the Plancherel decomposition for reductive groups. In [2] and [1] a notion of cusp forms was suggested for reductive symmetric spaces, specifically for hyperbolic spaces. This notion was adjusted in [14] to a notion for reductive symmetric spaces of split rank 1. ∗ Corresponding author.

E-mail addresses: [email protected] (M. Flensted-Jensen), [email protected] (J.J. Kuit). https://doi.org/10.1016/j.indag.2018.05.005 c 2018 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝

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The main problem one encounters when trying to define cusp forms, is convergence of the integrals involved. For a reductive symmetric space G/H of split rank 1, this problem was solved in [14] by identifying a class of parabolics subgroups P of G, the so-called h-compatible parabolic subgroups, for which the integrals ∫ φ(n) dn (0.1) N P /N P ∩H

are absolutely convergent for all Schwartz functions φ on G/H . In [16] it was shown that for the spaces SL(n, R)/GL(n − 1, R) the condition of h-compatibility is necessary for the integrals (0.1) to converge for all Schwartz functions φ. Cusp forms are then defined to be those Schwartz functions φ for which ∫ φ(gn) dn = 0 N P /N P ∩H

for every h-compatible parabolic subgroup P and every g ∈ G. For reductive symmetric spaces of split rank larger than 1, the methods in [15] cannot be used to show convergence of the integrals. For the spaces of type G/K ϵ , which are described in [11], the condition of h-compatibility is void. If σ denotes the involution determining the symmetric subgroup K ϵ , then the naive definition of cusp form involves the integrals (0.1) for all σ -parabolic subgroups P ̸= G, i.e., all parabolic subgroups P ̸= G such that σ (P) is opposite to P. This would require the integrals (0.1) to converge for all σ -parabolic subgroups P and all Schwartz functions φ. In this article we investigate the convergence of such integrals for three reductive symmetric spaces of split rank 2 of the type described in [11], namely SL(3, R)/SO(1, 2)e , SL(3, C)/SU(1, 2) and SL(3, H)/Sp(1, 2). For the first two of these spaces we show that for all σ -parabolic subgroups the integrals are absolutely convergent. We further show how one can characterize the different series of representation occurring in the Plancherel decomposition of these spaces with the use of these integrals. Using the higher-rank analogue of [14, Section 7.2] and a careful analysis of the residues occurring in the analogue of the formula in [14, Lemma 7.8] for the space SL(3, H)/Sp(1, 2), Erik van den Ban showed in unpublished notes that the integrals (0.1) are not converging for all minimal σ -parabolic subgroups of SL(3, H) and all Schwartz functions. We give a short and easy argument showing that not even for the maximal σ -parabolic subgroups all of the integrals are converging. The non-convergence of the integrals for this space raises the question whether it is possible to give a useful definition for cusp forms for reductive symmetric spaces of split rank larger than 1. The article is organized as follows. In Section 1 we describe the structure of the 3 above mentioned symmetric spaces, their Schwartz spaces and the parabolic subgroups. For the spaces SL(3, R)/SO(1, 2)e and SL(3, C)/SU(1, 2) we then show in Section 2 that all integrals are convergent and in Section 3 we make the connection with the Plancherel decompositions of these spaces. Finally, in Section 4 we prove that there exists a Schwartz-function on SL(3, H)/Sp(1, 2) such that the integrals (0.1) are divergent for some of the maximal and all of the minimal parabolic subgroups P. This paper grew out of discussions with Erik van den Ban and Henrik Schlichtkrull about explicit computations for a simple split rank 2 symmetric space. We want to thank both of them for their contribution through these discussions. In particular we want to thank Erik van den Ban for allowing us to publish our simple proof of his result on the non-convergence. The second author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) — 262362164. Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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1. Structure, parabolic subgroups and Schwartz spaces 1.1. Involutions Let F ∈ {R, C, H} and let G = SL(3, F). See [7, (1.120)] and the remark following that equation for the definition of SL(3, H). Let † denote conjugate transpose and let θ be the usual Cartan involution θ : g ↦→ (g −1 )† and let σ be the involution ⎛ ⎞ ⎛ 1 0 0 1 σ : g ↦→ ⎝0 −1 0 ⎠ θ (g) ⎝0 0 0 −1 0

0 −1 0

⎞ 0 0 ⎠. −1

We define H to be the connected open subgroup of G σ and K to be the maximal compact subgroup G θ . Note that • H = SO(1, 2)e and K = SO(3) if F = R; • H = SU(1, 2) and K = SU(3) if F = C; • H = Sp(1, 2) and K = Sp(3) if F = H. The involutions θ and σ commute. We use the same symbols for the involutions of g obtained by deriving θ and σ . Let p and q be the −1 eigenspaces of θ and σ respectively. Then g = k ⊕ p = h ⊕ q, where k and h are the Lie algebras of K and H respectively. 1.2. σ -stable maximal split abelian subalgebras Let b be a σ -stable maximal split abelian subalgebra of g. The split rank of G is equal to 2, while the split rank of H is equal to 1. Therefore dim(b) = 2 and dim(b ∩ h) ≤ 1. We define A to be the set of all σ -stable maximal split abelian subalgebras b such that dim(b ∩ h) = 0, i.e., b ⊂ q. Note that H acts on A. Proposition 1.1. The action of H on A is transitive. Proof. The proposition follows directly from [16, Lemma 1.1] and [9, Lemmas 4&7]. □ Let a be the Lie subalgebra of g consisting of all diagonal matrices. Then a ⊂ p ∩ q and thus a ∈ A. It follows from Proposition 1.1 that every maximal split abelian subalgebra in A is conjugate to a via an element in H . For later purposes we note here that the group N K ∩H (a)/Z K ∩H (a) consists of two elements: the equivalence class of the unit element and the equivalence class of ⎛ ⎞ 1 0 0 w 0 = ⎝ 0 0 1⎠ . (1.1) 0 −1 0

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1.3. H -conjugacy classes of minimal σ -parabolic subgroups Let Σ be the root system of a in g. For i ∈ {1, 2, 3} let ei ∈ a∗ be given by ( ) ei diag(x1 , x2 , x3 ) = xi . Then Σ = {ei − e j : 1 ≤ i, j ≤ 3, i ̸= j}. We write gi, j for the root space of a root ei − e j ∈ Σ . Each root space is linearly isomorphic to F. We define Σ1 := {e1 − e2 , e1 − e3 , e2 − e3 }, Σ2 := {e2 − e1 , e1 − e3 , e2 − e3 }, Σ3 := {e2 − e1 , e3 − e1 , e2 − e3 }.

(1.2)

Note that Σi , with i ∈ {1, 2, 3}, is a positive system of Σ . We define Pi to be the minimal parabolic subgroup such that a ⊂ Lie(Pi ) and the set of roots of Lie(Pi ) in a is equal to Σi . Proposition 1.2. Let P be a minimal parabolic subgroup of G. The following are equivalent. (i) (ii) (iii) (iv)

P is a σ -parabolic subgroup, i.e., σ P is opposite to P; P H is open in G; There exists a b ∈ A such that b ⊂ Lie(P); P is H -conjugate to one of the parabolic subgroups P1 , P2 or P3 .

Proof. (i) ⇒ (ii): Assume that σ P is opposite to P. Then 1+σ Lie(P) + h ⊇ Lie(P) + (Lie(P)) = Lie(P) + σ Lie(P) = g. (1.3) 2 This proves that P H is open in G. (ii) ⇒ (iii): Every element in A is a subspace that is contained in q, and vice versa, every maximal split abelian subspace of q is an element of A. The implication now follows from [12, Lemma 14]. (iii) ⇒ (iv): By Proposition 1.1 we may without loss of generality assume that a ⊂ Lie(P). Let k = w0 if g3,2 ⊂ Lie(P) (see (1.1)); otherwise, let k = e. Note that in both cases k ∈ N K ∩H (a). Therefore P ′ := k Pk −1 is a parabolic subgroup such that a ⊂ Lie(P ′ ). Moreover, g2,3 ⊂ Lie(P ′ ), hence P ′ is equal to one of the parabolic subgroups P1 , P2 or P3 . (iv) ⇒ (i): The parabolic subgroups Pi for i ∈ {1, 2, 3} are stable under the involution σ θ . Therefore they are all σ -parabolic subgroups. Any H -conjugate of a σ -parabolic subgroup is again a σ -parabolic subgroup, hence P is a σ -parabolic subgroup. □ Let Pσ be the set of all minimal σ -parabolic subgroups of G. For the symmetric spaces considered here the minimal σ -parabolic subgroups are in fact minimal parabolic subgroups. Note that H acts on Pσ . Proposition 1.3. The action of H on Pσ admits three orbits. Moreover, H \ Pσ = {[Pi ] : i = 1, 2, 3}, where [P] denotes the H -conjugacy class of P. Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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Proof. Let P ∈ Pσ . Since N K (a)/Z K (a) and N K ∩H (a)/Z K ∩H (a) consist of six and two elements, respectively, it follows from [12, Corollary 17] (see also [9, Proposition 1]) that there are three open H -orbits in G/P. In other words, there are three H -conjugacy classes of parabolic subgroups P ′ such that P ′ H is open in G. The proposition now follows from Proposition 1.2. □ We conclude this section with a relation between P1 and P3 . Recall w0 ∈ N K ∩H (a) from (1.1). Proposition 1.4. σ (w0 P1 w0−1 ) = P3 and σ (w0 P2 w0−1 ) = P2 . ( ) −1 Proof. Note that a ⊂ 0 ) σ Lie(P i ) for all i. Moreover, the sets of ( Lie(σ (w)0 Pi w0 )) = Ad(w ( ) roots of a in Ad(w0 ) σ Lie(P1 ) and Ad(w0 ) σ Lie(P2 ) are equal to Σ3 and Σ2 , respectively. This proves the proposition. □ 1.4. H -conjugacy classes of maximal σ -parabolic subgroups We define • • • •

Q1 Q2 Q3 Q4

to be the parabolic subgroup generated by to be the parabolic subgroup generated by to be the parabolic subgroup generated by to be the parabolic subgroup generated by

P1 and P2 , P3 and σ (P1 ), P1 and σ (P3 ), P2 and P3 .

For each i ∈ {1, 2, 3, 4} Q i is a maximal parabolic subgroup and the nilradical ni of Lie Q i is given by n1 = g1,3 ⊕ g2,3 ,

n2 = g2,1 ⊕ g3,1 ,

n3 = g1,2 ⊕ g1,3 ,

n4 = g2,1 ⊕ g2,3 .

Proposition 1.5. Let Q be a maximal parabolic subgroup. The following are equivalent. (i) (ii) (iii) (iv)

Q is a σ -parabolic subgroup, i.e., σ Q is opposite to Q; Q H is open in G; Q contains a minimal parabolic subgroup P such that P H is open; Q is H -conjugate to one of the parabolic subgroups Q 1 , Q 2 , Q 3 or Q 4 .

Proof. (i) ⇒ (ii): Assume that σ Q is opposite to Q. Then (1.3) holds with P replaced by Q and thus it follows that Q H is open in G. (ii) ⇒ (iii): Assume that Q H is open in G. Every minimal parabolic subgroup has only finitely many orbits in G/H . Therefore there exists a minimal parabolic subgroup P ⊂ Q such that P H is open. (iii) ⇒ (iv): Assume that Q contains a minimal parabolic subgroup P such that P H is open. From Proposition 1.2 it follows that Q is H -conjugate to a maximal parabolic subgroup containing one of the minimal parabolic subgroups P1 , P2 or P3 . Any maximal parabolic subgroup containing one of these three minimal parabolic subgroups is equal to Q 1 , Q 2 , Q 3 or Q 4 . (iv) ⇒ (i): The parabolic subgroups Q i for i ∈ {1, 2, 3, 4} are stable under the involution σ θ . Therefore they all are σ -parabolic subgroups. Any H -conjugate of a σ -parabolic subgroup is again a σ -parabolic subgroup, hence Q is a σ -parabolic subgroup. □ Let Qσ be the set of all maximal σ -parabolic subgroups of G. Note that H acts on Qσ . Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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Proposition 1.6. The action of H on Qσ admits four orbits. Moreover, H \ Qσ = {[Q i ] : i = 1, 2, 3, 4}, where [Q] denotes the H -conjugacy class of Q. Proof. There are two G-conjugacy classes of maximal parabolic subgroups. Let [Q]G denote the G-conjugacy class of Q. Then [Q 1 ]G = [Q 2 ]G ̸= [Q 3 ]G = [Q 4 ]G . From [12, Corollary 16 (2)] it easily seen that Q i admits two open orbits in G/H for i ∈ {1, 2, 3, 4}. (From the proof of the corollary it follows that one may take the group A in the theorem to be equal to exp(a).) In view of Proposition 1.5 it follows that there are two H -conjugacy classes of σ -parabolic subgroups in each G-conjugacy class [Q i ]G , hence in total there are four H -conjugacy classes in Qσ . The remaining claim follows from Proposition 1.5. □ Recall w0 ∈ N K ∩H (a) from (1.1). Proposition 1.7. σ Q 2 = Q 3 and σ (w0 Q 1 w0−1 ) = Q 4 . Proof. The identities follow directly from the definition of the groups Q i and Proposition 1.4. □ 1.5. Polar decomposition, the Schwartz space and tempered functions In this section we discuss the polar decomposition for G/H and some sub-symmetric spaces of G/H . We further give a definition of Harish-Chandra Schwartz functions and tempered functions. Let L be a σ -stable closed subgroup of G. Assume that L is a reductive Lie group. Let θ L be a Cartan involution of L that commutes with σ and let K L = L θ L be the corresponding σ -stable maximal compact subgroup of L. Let a L be a maximal split abelian subalgebra of l contained in l ∩ q and let A L = exp(a L ). Finally let HL be the symmetric subgroup L σ = H ∩ L of L. (Note that L = G, K L = K , A L = A and HL = H are valid choices for the above defined subgroups.) The space L/HL admits a polar decomposition: the map K L × A L → L/HL ;

(k, a) ↦→ ka · HL

is surjective. Moreover, if a · HL ∈ K L a ′ · HL with a, a ′ ∈ A L , then there exists an element k ∈ N K L ∩HL (a L ) such that a = ka ′ k −1 . Let PL be a minimal σ -parabolic subgroup of L. We define ρ PL ∈ a∗L by ⏐ ) 1 ( (Y ∈ a L ). ρ PL (Y ) := tr ad(Y )⏐n PL 2 Let W L be the Weyl group of the root system in a L . Definition 1.8. A Schwartz function on L/HL is a smooth function φ : L/HL → C, such that for every u ∈ U(l) and r ≥ 0 the seminorm (∑ )( ⏐ )r ⏐ L µu,r (φ) := sup sup a w·ρ PL 1 + ∥ log(a)∥ ⏐(uφ)(ka · HL )⏐ k∈K L a∈A L w∈W L

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is finite. Here the action of U(l) on C ∞ (L/HL ) is obtained from the left-regular representation of L on C ∞ (L/HL ). We denote the vector space of Schwartz functions on L/HL by C(L/HL ) and equip C(L/HL ) with the topology induced by the mentioned seminorms. Our definition of Schwartz functions is equivalent to the one in [13, section 17]. ∞ We further define Ctemp (L/HL ) to be the space of smooth functions on L/HL which are tempered as distributions on L/HL , i.e., belong to the dual C ′ (L/HL ) of the Schwartz space ∞ C(L/HL ). We equip the space Ctemp (L/HL ) with the coarsest locally convex topology such that the inclusion maps into C ∞ (L/HL ) and C ′ (L/HL ) are both continuous. Here C ∞ (L/HL ) is equipped with the usual Fr´echet topology and C ′ (L/HL ) is equipped with the strong dual topology. We finish this section with a more precise description of C(G/H ). We choose θG = θ and aG = a. Define Φ : G → R>0 by Φ(g) = ∥gσ (g)−1 ∥2H S ∥σ (g)g −1 ∥2H S

(g ∈ G),

(1.4)

where ∥ · ∥ H S denotes the Hilbert–Schmidt norm on Mat(n, F). We further define V := {t ∈ R3 :

3 ∑

ti = 0}.

i=1

For t ∈ V we further define ⎛ t ⎞ e1 0 0 at := ⎝ 0 et2 0 ⎠ . 0 0 e t3

(1.5)

Lemma 1.9. Let g ∈ G and t ∈ V . If g ∈ K at · H , then Φ(g) =

3 (∑ i=1

e

4ti

3 )(∑

e−4ti

)

(1.6)

i=1

( ) ( ) ( ) = 3 + 2 cosh 4(t1 − t2 ) + 2 cosh 4(t1 − t3 ) + 2 cosh 4(t2 − t3 ) . In particular, Φ is left K -invariant, right H -invariant and Φ ◦σ = Φ. Moreover, Φ| A is invariant under the action of the Weyl group of the root system Σ in a. Proof. A straightforward computation shows that ( ( )t ) ∥gσ (g)−1 ∥2H S = tr gσ (g)−1 gσ (g)−1 = tr (at4 ) and ( ( )t ) ∥σ (g)g −1 ∥2H S = tr σ (g)g −1 σ (g)g −1 = tr (at−4 ). The equalities in (1.6) follow from (1.5). Eq. (1.6) may be rewritten as ∑ ( ) Φ(ka · H ) = 3 + cosh 4α(log a) (k ∈ K , a ∈ A). α∈Σ

From this identity the claimed invariances are clear. □ Remark 1.10. From (1.6) it follows that Φ(g) ≥ 9 for all g ∈ G. Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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Let k = dimR F. Let ρ1 be half the sum of the roots in Σ1 , see (1.2), and let x ∈ a be in the corresponding positive Weyl chamber. In view of Lemma 1.9 4

4

3 + e k ρ1 x ≤ Φ(exp x) ≤ 3 + 3e k ρ1 x . ⏐ The Weyl group invariance of Φ ⏐ A now implies that a smooth function φ : G/H → C belongs to the Schwartz space C(G/H ) if and only if for every u ∈ U(g) and r ≥ 0 the seminorm ⏐ )r ⏐ k( µu,r (φ) := sup Φ(x) 4 log ◦Φ(x) ⏐(uφ)(x)⏐ x∈G/H

is finite. The topology induced by the seminorms µu,r is equal to the one induced from the G seminorms µu,r . Note that N Q 1 = {n y,z : y, z ∈ F}, where ⎞ ⎛ 1 0 z (y, z ∈ F). (1.7) n y,z := ⎝0 1 y ⎠ 0 0 1 The following lemma follows from a straightforward computation of the right-hand side of (1.4). Lemma 1.11. Let t ∈ V and y, z ∈ F. Then Φ(at n y,z ) ( ) = e4t1 (1 − |z|2 )2 + e4t2 (1 + |y|2 )2 + e4t3 + 2e−2t1 |y|2 + 2e−2t2 |z|2 + 2e−2t3 |y|2 |z|2 ( ) × e−4t1 + e−4t2 + e−4t3 (1 + |y|2 − |z|2 )2 + 2e2t1 |y|2 + 2e2t2 |z|2 . 2. Convergence of cuspidal integrals Throughout this section, let F = R or F = C. 2.1. Main theorem Theorem 2.1. Let P be a σ -parabolic subgroup and let P = M P A P N P be a Langlands decomposition of P so that M P and A P are σ -stable. We set L P := M P A P = P ∩ σ (P). (i) For every φ ∈ C(G/H ) and g ∈ G the integral ∫ φ(gn) dn R P φ(g) :=

(2.1)

NP

is absolutely convergent and the function R P φ thus obtained is a smooth function on G/(L P ∩ H )N P . (ii) Let δ P be the character on P given by 1 ⏐ ⏐ 1 δ P (l) := |Ad(l)⏐Lie(P) | 2 = |Ad(l)⏐n | 2 (l ∈ L P ). P

Define for φ ∈ C(G/H ) the function H P φ ∈ C ∞ (L P ) by H P φ(l) := δ P (l)R P φ(l)

(l ∈ L P ).

Then H P( φ is right L P)∩ H -invariant and H P defines a continuous linear map from C(G/H ) ∞ to Ctemp L P /L P ∩ H . Moreover, Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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a. if F = R, then H P defines a continuous linear map from C(G/H ) to C L P /L P ∩ H ; ⏐ b. if F = C, then φ ↦→ H P φ ⏐ M defines a continuous linear map from C(G/H ) to P ( ) C M P /M P ∩ H . (

)

In the remainder of Section 2 we give the proof for Theorem 2.1. 2.2. Some estimates We define the functions M : R → [9, ∞);

x ↦→ max(9, x)

and L := log ◦M : R → [log 9, ∞). Note that M and L are monotonically increasing. Lemma 2.2. Let r > 2, r1 ≥ 2 and r2 ≥ 0 and assume that r = r1 + r2 . Then there exists a c > 0 such that for all κ1 , κ2 > 0 the following hold. ∫∞ ( )−r )− 1 ( −1 ds ≤ cκ1 4 L(κ1 )−r1 +1 L(κ2 )−r2 . (i) 0 M κ1 (s 2 ± 1)2 + κ2 4 L κ1 (s 2 ± 1)2 + κ2 ∫∞ )−r −1 ( (ii) 0 |s 2 ± 1| 2 L κ1 (s 2 ± 1)2 ds ≤ cL(κ1−1 )L(κ1 )−r +1 . ∫∞ ( 2 )−r )− 1 ( 2 −1 ds ≤ cκ1 2 L(κ2 )−r +2 . (iii) 0 M κ1 s + κ2 2 L κ1 s + κ2 ∫ ∞( )−r )− 1 ( −1 (iv) 0 κ1 s 2 + κ2 2 L κ1 s 2 + κ2 ds ≤ cκ1 2 L(κ2−1 )L(κ2 )−r +2 . The estimates in (i) and (iii) also hold with κ2 = 0. Proof. In order to prove (i) and (ii), it suffices to consider the desired inequalities only for the case with the minus signs in the integrand. We start with (i) and first note that integral is smaller than or equal to L(κ2 )−r2 I (κ1 ), where ∫ ∞ )−r ( )− 1 ( I (κ1 ) := M κ1 (s 2 − 1)2 4 L κ1 (s 2 − 1)2 1 ds 0 ∫ )− 1 ( )− 1 ( )−r 1 ∞( = t + 1 2 M κ1 t 2 4 L κ1 t 2 1 dt. 2 −1 Note that the integral in I (κ1 ) is absolutely convergent for every κ1 > 0 and the resulting function I is continuous. We need to prove that −1

I (κ1 ) ≤ cκ1 4 L(κ1 )−r1 +1 for some c > 0. It is enough to consider small and large κ1 . First assume that κ1 ≤ 9. Then there exist c, c′ , c′′ > 0 such that ∫ )− 1 ( )− 1 ( )−2 1 ∞( I (κ1 ) ≤ t + 1 2 M κ1 t 2 4 L κ1 t 2 dt 2 −1 ∫ √3 ∫ κ1 ( )− 1 )− 1 1 ( )−2 1 −1 ∞ ( =c t + 1 2 dt + κ1 4 t + 1 2 t − 2 log κ1 t 2 dt 2 √3 −1 κ1 ∫ ∞ −1 −1 ≤ c′ κ1 4 + c′ κ1 4 u −2 du log(9)

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−1

= c′′ κ1 4 . −1

1

Now assume κ1 ≥ 93 . We define δ = κ1 3 . Then δ ≤ 9−1 and κ1 t 2 ≤ κ13 if and only if |t| ≤ δ. 1

Therefore, |t| ≥ δ implies κ1 t 2 ≥ κ13 ≥ 9 and we find that there exist c, c′ > 0 such that ∫ δ 1 I (κ1 ) ≤ c |1 − t|− 2 dt 0 ∫ 1 1 1 1 − 14 3 −r1 |1 − t|− 2 t − 2 dt + 2κ1 log(κ1 ) δ ∫ ∞ − 41 − 21 − 12 + κ1 (1 + t) t log(κ1 t 2 )−r1 dt ≤

1 1 − c′ δ + c′ κ1 4 log(κ1 )−r1 ∫ ∞ − 41 + κ1 t −1 log(κ1 t 2 )−r1 1

dt.

−1

The latter is smaller than c′′ κ1 4 L(κ1 )−r1 +1 for some c′′ > 0 as ∫ ∞ ∫ 1 ∞ 1 t −1 log(κ1 t 2 )−r1 dt = s −r1 ds = log(κ1 )−r1 +1 . 2 2(r − 1) 1 1 log(κ1 ) This proves (i). −1

We move on to (ii). Let δ := 3κ1 2 . Since |s 2 − 1| ≤ δ if and only if κ1 (s 2 − 1)2 ≤ 9 we have ∫ ∞ )−r −1 ( |s 2 − 1| 2 L κ1 (s 2 − 1)2 ds 0 ∫ 1 )−r ( )− 1 ( ds = κ14 M κ1 (s 2 − 1)2 4 L κ1 (s 2 − 1)2 {s∈[0,∞):|s 2 −1|≥δ} ∫ −1 |s 2 − 1| 2 ds +c (2.2) {s∈[0,∞):|s 2 −1|≤δ}

with c = log(9) . In view of (i) with κ2 = 0 the first term on the right hand side of (2.2) is smaller than or equal to c′ L(κ1 )−r +1 for some c′ > 0. Now we turn our attention to the integral in the second term on the right-hand side of (2.2). Up to a constant it is equal to ∫ δ 1 1 J (κ1 ) := (t + 1)− 2 |t|− 2 dt. −r

− min(1,δ)

Note that the integral is absolutely convergent and that the function J is continuous. It suffices to prove that J (κ1 ) ≤ cL(κ1−1 )L(κ1 )−r +1 for some c > 0. It is enough to consider small and large κ1 . First let κ1 ≤ 9. Then there exists a c > 0 such that ∫ δ 1 1 J (κ1 ) = (t + 1)− 2 |t|− 2 dt ≤ cL(κ1−1 ). −1

Next, let κ1 ≫ 9. Then there exists a c > 0 such that ∫ δ ∫ δ 1 1 1 −1 J (κ1 ) = (t + 1)− 2 |t|− 2 dt ≤ 2 |t|− 2 dt ≤ cκ1 4 . −δ

−δ

This proves (ii). Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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√ By performing a substitution of variables s ′ = κ1 s we may ( reduce ) the proof of (iii) and (iv) to the case that κ1 = 1. Since L is an increasing function, L s 2 + κ2 ≥ L(κ2 ), hence ( )−r ( )−r +2 ( 2 )−2 L s 2 + κ2 ≤ L κ2 L s + κ2 . Using that the integrand decreases as a function of κ2 we find ∫ ∞ ∫ ∞ ( 2 )− 1 ( 2 )−2 1 2 M s + κ2 L s + κ2 ds ≤ M(s 2 )− 2 L(s 2 )−2 ds < ∞. 0

0

This proves (iii). To prove (iv) it suffices to show that ∫ ∞ )−2 ( 2 )− 1 ( ds ≤ cL(κ2−1 ) s + κ2 2 L s 2 + κ2 0

for some c > 0. Now the integral on the left-hand side is equal to ∫ ∞ ∫ 1 ( 2 )− 1 ( ( 2 )− 1 ( )−2 )−2 s + κ2 2 L s 2 + κ2 s + κ2 2 L s 2 + κ2 ds + ds. 1

0

The second term is bounded by ∫ ∞ ( )−2 |s|−1 L s 2 ds < ∞; 1

the first term is smaller than or equal to 1

1(

∫ c

s 2 + κ2

)− 1

2

− κ2 2

∫ ds = c

0

0

(

)− 1 −1 s 2 + 1 2 ds = c arsinh(κ2 2 )

−2

with c = log(9) . This proves (iv) as arsinh(x) ∼ log(x) for x → ∞. □ Proposition 2.3. Let r > 4. (i) Assume that F = R. There exists a c > 0 such that for every t ∈ V ∫ )−r ρQ 1( at 1 Φ(at n)− 4 log ◦Φ(at n) dn ≤ c cosh(t1 − t2 )−1 (1 + ∥t∥)−r +4 N Q1

and ρQ3

∫

)−r 1( Φ(at n)− 4 log ◦Φ(at n) dn ≤ c cosh(t2 − t3 )−1 (1 + ∥t∥)−r +4 .

at

N Q3

(ii) Assume that F = C. There exist c, C > 0 such that for every t ∈ V ∫ )−r ρQ 1( at 1 Φ(at n)− 2 log ◦Φ(at n) dn N Q1

)−r +4 ( ≤ c cosh(t1 − t2 )−1 L(e−t3 )L e3t3 cosh(t1 − t2 ) ( )−r +4 ≤ C L(e−t3 )r −3 cosh(t1 − t2 )−1 1 + |t1 − t2 |

(2.3) (2.4)

and ρQ3

∫

)−r 1( Φ(at n)− 2 log ◦Φ(at n) dn

at

N Q3

( )−r +4 ≤ c cosh(t2 − t3 )−1 L(et1 )L e−3t1 cosh(t2 − t3 ) ( )−r +4 ≤ C L(et1 )r −3 cosh(t2 − t3 )−1 1 + |t2 − t3 | .

Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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Proof. We will prove the estimates for the parabolic subgroup Q 1 ; the proof for the estimates for Q 3 is similar. Let k = dimR (F). Let t ∈ V and define ∫ )−r ρQ k( It := at 1 Φ(at n)− 4 log ◦Φ(at n) dn. N Q1

Then It = e k

t1 +t2 −2t3 2

∫ ∫ F

)−r k( Φ(at n y,z )− 4 log ◦Φ(at n y,z ) dz dy,

F

where n y,z is given by (1.7). Since Φ(x) ≥ 9 for all x ∈ G, see Remark 1.10, we have log ◦Φ = L ◦ Φ. Define ( )( ) Φ1 (t, y, z) = e4t2 (1 + |y|2 )2 + e4t3 e−4t1 + e−4t2 + e−4t3 (1 + |y|2 − |z|2 )2 ( )( ) Φ2 (t, y, z) = e4t1 (1 − |z|2 )2 + e4t3 e−4t1 + e−4t2 + e−4t3 (1 + |y|2 − |z|2 )2 . It follows from Lemma 1.11 that Φ(at n y,z ) can be estimated from below by both Φ1 (t, y, z) and Φ2 (t, y, z). Now assume that F = R. √ We first use the estimate Φ(at n y,z ) ≥ Φ1 (t, y, z). We perform the substitution of variables z = y 2 + 1v and thus obtain that there exists a constant c1 > 0 such that ∫ ∫ t1 +t2 −2t3 )−r 1 ( M(Φ1 (t, y, z))− 4 L Φ1 (t, y, z) dz dy It ≤ e 2 ∫R ∞R∫ ∞ t1 +t2 −2t3 1 (y 2 + 1) 2 = c1 e 2 0

0

)( )]− 41 × M e4t2 (y 2 + 1)2 + e4t3 e−4t1 + e−4t2 + e−4t3 (y 2 + 1)2 (v 2 − 1)2 [( )( )]−r dv dy. × L e4t2 (y 2 + 1)2 + e4t3 e−4t1 + e−4t2 + e−4t3 (y 2 + 1)2 (v 2 − 1)2 [(

We apply Lemma 2.2(i) to the inner integral with ( ) ( )( ) κ1 = e4(t2 −t3 ) (y 2 + 1)2 + 1 (y 2 + 1)2 , κ2 = e4t2 (y 2 + 1)2 + e4t3 e−4t1 + e−4t2 , and thus we see that for every 2 ≤ r1 ≤ r and r2 = r − r1 there exists a constant c2 > 0 such that It is smaller than or equal to ∫ ∞ )−r1 +1 t1 +t2 −2t3 ( 4(t −t ) 2 )− 1 (( ) c2 e 2 e 2 3 (y + 1)2 + 1 4 L e4(t2 −t3 ) (y 2 + 1)2 + 1 (y 2 + 1)2 0 (( )( ))−r2 dy. × L e4t2 (y 2 + 1)2 + e4t3 e−4t1 + e−4t2 We now first put r1 = r and r2 = 0. Using that L(x 2 ) ∼ L(x) for x → ∞, we obtain that there exists a c3 > 0 such that ∫ ∞ ( )−r +1 t1 −t2 1 I t ≤ c3 e 2 (y 2 + 1)− 2 L e2(t2 −t3 ) (y 2 + 1)2 dy. 0

By Lemma 2.2(ii) there exists a c4 > 0 such that t1 −t2 ( ) ( )−r +2 It ≤ c4 e 2 L e2(t3 −t2 ) L e2(t2 −t3 ) . Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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Secondly we take r1 = 2 and r2 = r − 2 and use that ( 4t 2 )( ) ( ) e 2 (y + 1)2 + e4t3 e−4t1 + e−4t2 ≥ max e4(t3 −t1 ) + e4(t3 −t2 ) , (y 2 + 1)2 . This yields the existence of constants c5 , c6 > 0 such that It is smaller than or equal to ( )−r +4 ∫ ∞ ( )−2 t1 +t2 −2t3 )− 1 ( e4(t2 −t3 ) (y 2 + 1)2 + 1 4 L (y 2 + 1)2 dy c5 e 2 L e4(t3 −t1 ) + e4(t3 −t2 ) 0 ( )−r +4 t1 −t2 ≤ c6 e 2 L e4(t3 −t1 ) + e4(t3 −t2 ) . )−1 (( ) in the integrand by a (Here we have estimated a factor of L e4(t2 −t3 ) (y 2 + 1)2 + 1 (y 2 + 1)2 constant.) These two inequalities for It imply that for every closed cone Γ with the property that Γ \ {0} ⊆ {t ∈ V : t2 − t3 > 0} ∪ {t ∈ V : t3 − t1 > 0} ∪ {t ∈ V : t3 − t2 > 0},

(2.5)

there exists a constant c > 0 so that It ≤ ce

t1 −t2 2

(1 + ∥t∥)−r +4

(t ∈ Γ ).

The cone Γ = {t ∈ V : t2 ≥ t1 } is equal to the union Γ = {t ∈ V : t3 ≥ t2 ≥ t1 } ∪ {t ∈ V : t2 ≥ t3 ≥ t1 } ∪ {t ∈ V : t2 ≥ t1 ≥ t3 }. Using that t1 = t2 = t3 and t ∈ V implies that t = 0, it follows that Γ satisfies (2.5). Therefore there exists a c > 0 so that for every t ∈ V with t2 ≥ t1 It ≤ ce

t1 −t2 2

(1 + ∥t∥)−r +4 .

We √ now use the estimate Φ(at n y,z ) ≥ Φ2 (t, y, z). We perform the substitution of variables y = |z 2 − 1|v and thus we obtain that there exists a constant c1 > 0 such that It is smaller than or equal to ∫ ∞∫ ∞ 1 t1 +t2 −2t3 2 |z 2 − 1| 2 c1 e 0

0

)( )]− 41 × M e4t1 (1 − z 2 )2 + e4t3 e−4t1 + e−4t2 + e−4t3 (z 2 − 1)2 (1 − v 2 )2 [( )( )]−r × L e4t1 (1 − z 2 )2 + e4t3 e−4t1 + e−4t2 + e−4t3 (z 2 − 1)2 (1 − v 2 )2 dv dz. [(

We apply Lemma 2.2(i) to the inner integral with )( ) ( ) ( κ1 = e4(t1 −t3 ) (1 − z 2 )2 + 1 (z 2 − 1)2 , κ2 = e4t1 (1 − z 2 )2 + e4t3 e−4t1 + e−4t2 , and thus we see that for every 2 ≤ r1 ≤ r and r2 = r − r1 there exists a constant c2 > 0 such that It is smaller than or equal to ∫ ∞ )−r1 +1 t1 +t2 −2t3 ( 4(t −t ) )− 1 ( ( ) c2 e 2 e 1 3 (1 − z 2 )2 + 1 4 L e4(t1 −t3 ) (1 − z 2 )2 + 1 (z 2 − 1)2 0 (( )( ))−r2 × L e4t1 (1 − z 2 )2 + e4t3 e−4t1 + e−4t2 dz. Applying Lemma 2.2(ii) to the remaining integral as above, we obtain a constant c3 > 0 such t1 −t2 that e 2 It is smaller than or equal to ( ( )−r +4 ) c3 min L(e2(t3 −t1 ) )L(e2(t1 −t3 ) )−r +2 , L e4(t3 −t1 ) + e4(t3 −t2 ) . Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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It follows as before that there exists a c > 0 such that for every t ∈ V with t1 ≥ t2 It ≤ ce

t2 −t1 2

(1 + ∥t∥)−r +4 .

This proves (i). Next, assume that F = C. We first use the estimate Φ(at n y,z ) ≥ Φ1 (t, y, z). After introducing polar coordinates and subsequently performing the substitution of variables v = |z|2 − |y|2 − 1, w = |y|2 + 1, we obtain that there exists a constant c1 > 0 such that ∫ ∫ )−r 1 ( M(Φ1 (t, y, z))− 2 L Φ1 (t, y, z) dz dy It ≤ et1 +t2 −2t3 C C ∫ ∞∫ ∞ [ ( )( )]− 12 = c1 et1 +t2 −2t3 M e4t2 w2 + e4t3 e−4t1 + e−4t2 + e−4t3 v 2 1 −w [( )( )]−r dv dw × L e4t2 w2 + e4t3 e−4t1 + e−4t2 + e−4t3 v 2 ∫ ∞∫ ∞ [ ( )( )]− 21 ≤ 2c1 et1 +t2 −2t3 M e4t2 w 2 + e4t3 e−4t1 + e−4t2 + e−4t3 v 2 0 0 [( )( )]−r dw dv. × L e4t2 w2 + e4t3 e−4t1 + e−4t2 + e−4t3 v 2 We apply Lemma 2.2(iii) to the inner integral with κ1 = e4(t2 −t1 ) + 1 + e4(t2 −t3 ) v 2 and κ2 = e4(t3 −t1 ) + e4(t3 −t2 ) + v 2 and we thus find that there exists a constant c2 > 0 such that ∫ ∞ ( 4(t −t ) )− 1 ( )−r +2 t1 +t2 −2t3 I t ≤ c2 e e 2 1 + 1 + e4(t2 −t3 ) v 2 2 L e4(t3 −t1 ) + e4(t3 −t2 ) + v 2 dv ∫ ∞0 )−r +2 ( 4(t −t ) )− 1 ( = c2 et1 −t2 dv. e 3 1 + e4(t3 −t2 ) + v 2 2 L e4(t3 −t1 ) + e4(t3 −t2 ) + v 2 0

We may now apply Lemma 2.2(iv) to the remaining integral with κ1 = 1 and κ2 = e4(t3 −t1 ) + e4(t3 −t2 ) . It follows that there exists a c > 0 such that ( ) ( )−r +4 It ≤ cet1 −t2 L (e4(t3 −t1 ) + e4(t3 −t2 ) )−1 L e4(t3 −t1 ) + e4(t3 −t2 ) . Using the estimate Φ(at n y,z ) ≥ Φ2 (t, y, z) and a similar computation we obtain that there exists a constant c > 0 such that ( ) ( )−r +4 It ≤ cet2 −t1 L (e4(t3 −t1 ) + e4(t3 −t2 ) )−1 L e4(t3 −t1 ) + e4(t3 −t2 ) . Now observe that )−r +4 ( ) ( L (e4(t3 −t1 ) + e4(t3 −t2 ) )−1 L e4(t3 −t1 ) + e4(t3 −t2 ) )−r +4 (( )−1 ) ( 6t = L 2e6t3 cosh(2t1 − 2t2 ) L 2e 3 cosh(2t1 − 2t2 ) ( ) ( )−r +4 ≤ L e−6t3 L 2e6t3 cosh(2t1 − 2t2 ) ( ) ( )−r +4 ≤ cL e−t3 L e3t3 cosh(t1 − t2 ) for some c > 0. This proves the estimate (2.3). For a, b > 0 { log(9), (b ≤ 9a) −1 L(a b) = − log(a) + log(b), (b ≥ 9a). We claim that for sufficiently large a > 0 1 + log(b) L(a −1 b) ≥ (b > 0). 2 log(a) Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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If a > 0 is large and b ≤ 9a then 1 + log(b) 1 + log(9a) ≤ < log(9). 2 log(a) 2 log(a) is strictly smaller than the Moreover, if a > 0 is large, then the slope of the function x ↦→ 2 1+x log(a) slope of x ↦→ − log(a) + x. This proves the claim. It follows that ( ) ( −1 ) ( ) 1 + log cosh(v) 1 + |v| −1 L u cosh(v) ≥ L M(u) cosh(v) ≥ ≥ . 2L(u) 2L(u) The second inequality (2.4) follows from these estimates. □ 2.3. Proof of Theorem 2.1 for maximal σ -parabolic subgroups In this section we give the proof for Theorem 2.1 for a maximal σ -parabolic subgroup, which we here denote Q. We write L Q for the σ -stable Levi subgroup of Q, i.e., L Q = σ (Q)∩ Q. Let P be a minimal σ -parabolic subgroup contained in Q and let Q = M Q A Q N Q and P = M P A P N P be Langlands decompositions of Q and P respectively, such that A P and A Q are σ -stable and A Q ⊆ A P . Let R := M Q ∩ P. Then R is a minimal σ -parabolic subgroup of M Q . We first list a number of conclusions that can be drawn from the calculations in Section 2.2. Lemma 2.4. 1a. If F = R, then for every r > 4 there exists a constant c > 0 such that for every a ∈ A P ∫ ( ∑ )−1 ( )−r )−r +4 1( a w·ρ R 1 + ∥ log(a)∥ . aρQ Φ(an)− 4 log Φ(an) dn ≤ c NQ

w∈W M Q

1b. If F = C, then for every r > 4 there exists a constant c > 0 such that for every a ∈ A Q and b ∈ A P ∩ M Q ∫ )−r 1( ρQ Φ(abn)− 2 log Φ(abn) dn a NQ

≤c

( ∑

bw·ρ R

)−1 (

)−r +4 1 + ∥ log(b)∥ ∥ log a∥r −3 .

w∈W M Q

2. There exists a constant C > 0 such that for every φ ∈ C(G/H ) ⏐ ⏐ ⏐ ⏐ sup sup ⏐(1 + ∥ log(a)∥)−2 a ρ Q R Q φ(ka)⏐ ≤ Cµ1,5 (φ), k∈K a∈A

3a. If F = R, then for every r > 0 there exists a constant C > 0 such that for every φ ∈ C(G/H ) ) L ( µ1,rQ H Q φ ≤ Cµ1,r +4 (φ), 3b. If F = C, then for every r > 0 there exists a constant C > 0 such that for every φ ∈ C(G/H ) ( )) M ( sup (1 + ∥ log(a)∥)−r −1 µ1,rQ H Q φ(a · ) ≤ Cµ1,r +4 (φ), a∈A Q

Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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Proof. It suffices to prove the lemma for only one parabolic subgroup in each H -conjugacy class of maximal σ -parabolic subgroups, hence by Proposition 1.6 we may assume that Q = Q i for some i ∈ {1, 2, 3, 4}. Since Φ is invariant under composition with σ and conjugation with w0 , it is in view of Proposition 1.7 enough to prove the theorem for Q = Q 1 and Q = Q 3 . The assertions in 1. now follow from Proposition 2.3. The claims in 2. and 3. follow directly from the estimates in 1. □ We now give the proof of Theorem 2.1 for maximal parabolic subgroups Q. Since N Q H/H is closed in G/H , the Radon transform R Q defines a continuous linear map Cc∞ (G/H ) → C(G/(L Q ∩ H )N Q ). It follows from 1a and 1b in Lemma 2.4 that this map extends to a continuous linear map R Q : C(G/H ) → C(G/(L Q ∩ H )N Q ) which for all φ ∈ C(G/H ) and g ∈ G is given by (2.1) with absolutely convergent integrals. Since R Q is G-equivariant and the left regular representation of G on C(G/H ) is smooth, R Q in fact defines a continuous linear map C(G/H ) → C ∞ (G/(L Q ∩ H )N Q ). Moreover, for every φ ∈ C(G/H ) and u ∈ U(g) u(R Q φ) = R Q (uφ). Since δ Q is a smooth and L Q ∩ H invariant character on L Q , H Q is a continuous linear map C(G/H ) → C ∞ (L Q /L Q ∩ H ). It follows from 2 in Lemma 2.4 that H Q also defines a continuous linear map C(G/H ) → C(L Q /L Q ∩ H )′ and thus H Q is in fact a continuous map to ∞ Ctemp (L Q /L Q ∩ H ). Let φ ∈ C(G/H ) and let u ∈ U(l Q ). It follows from the Leibniz rule and the fact that δ Q is a character on L Q , that there exists a v ∈ U(l Q ) such that ⏐ ) ⏐ ⏐ ( u(H Q φ) = u δ Q (R Q φ)⏐ L = δ Q v(R Q φ)⏐ L = δ Q R Q (vφ)⏐ L = H Q (vφ). Q

Q

Q

If F = R, then 3a in Lemma 2.4 implies that ) ) ) L ( L ( L ( µu,rQ H Q φ = µ1,rQ uH Q φ = µ1,rQ H Q (vφ) ≤ Cµ1,4+r (vφ) = Cµv,4+r (φ). ( ) This proves that H Q defines a continuous linear map C(G/H ) → C L⏐Q /(L Q ∩ H ) . If F = C, the same argument, with L Q replaced by M Q shows that φ ↦→ H Q φ ⏐ M defines a continuous Q ( ) linear map C(G/H ) → C M Q /(M Q ∩ H ) . 2.4. Proof of Theorem 2.1 for minimal σ -parabolic subgroups In this section we give the proof for Theorem 2.1 in case P is a minimal σ -parabolic subgroup. Let P be a minimal σ -parabolic subgroup and let Q be a maximal σ -parabolic subgroup containing P. Let P = M P A P N P and Q = M Q A Q N Q be Langlands decompositions so that A P and A Q are σ -stable and A Q ⊆ A P . Then M Q ∩ H is a symmetric subgroup of M Q . The group R := P ∩ M Q is a minimal σ -parabolic subgroup of M Q . Note that M Q is isomorphic to the group {g ∈ GL(2, F) : |det g| = 1}. We differentiate between three cases. (a) Q is H -conjugate to Q 2 or Q 3 . Then M Q ∩ H is a maximal compact subgroup of M Q and hence it is isomorphic to O(2) if F = R and U(2) if F = C. Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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(b) F = R and Q is H -conjugate to Q 1 or Q 4 . Then M Q ∩ H is isomorphic to O(1, 1). (c) F = C and Q is H -conjugate to Q 1 or Q 4 . Then M Q ∩ H is isomorphic to U(1, 1). In this case M Q /M Q ∩ H ≃ PSL(2, C)/PSU(1, 1) ≃ SO(3, 1)e /SO(2, 1)e . as SO(3, 1)e -homogeneous spaces. Note that the Schwartz space and the Radon transforms for M Q /M Q ∩ H considered as a homogeneous space for SO(3, 1)e coincide with the Schwartz space and the Radon transforms for M Q /M Q ∩ H considered as a homogeneous space for M Q . For case (a) it is well known and for (b) and (c) it follows from [1, Theorem 5.1] that the integral ∫ φ(n) dn NR

is absolutely convergent for every continuous function φ : M Q /M Q ∩ H → C satisfying sup

(a ρ R + a −ρ R )(1 + ∥ log a∥)|φ(ka)| < ∞.

sup

k∈K ∩M Q a∈A P ∩M Q

Moreover, if r > 1 and φ satisfies sup

(a ρ R + a −ρ R )(1 + ∥ log a∥)r |φ(ka)| < ∞,

sup

k∈K ∩M Q a∈A P ∩M Q

then the function R R φ(g) :=

∫

φ(gn) dn

(g ∈ M Q )

NR

satisfies in the cases (a) and (b) sup

sup

(1 + ∥ log a∥)r −1 a ρ R R R φ(ka) < ∞.

k∈K ∩M Q a∈A P ∩M Q

In case (c) such strong estimates do not hold, but we still have sup

sup

a ρ R R R φ(ka) < ∞.

k∈K ∩M Q a∈A P ∩M Q

These estimates are well known if M Q ∩ H is a maximal compact subgroup of M Q and in the cases (b) and (c) they again follow from [1, Theorem 5.1]. In all cases the multiplication map (ν, n) → νn

NR × NQ → NP ;

is a diffeomorphism with Jacobian equal to the constant function 1. Therefore ∫ ∫ ∫ ψ(νn) dn dν ψ(gn) dn = NP

NR

(2.6)

NQ

for every ψ ⏐∈ L 1 (N P ). Since δ Q ⏐ N = 1, it follows from 1a and 1b in Lemma 2.4 that for r > 5 and all g ∈ G the R integral ∫ )−r k( Φ(gn)− 4 log Φ(gn) dn NP

is absolutely convergent. Moreover, we have the following estimates. Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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(i) If F = R, then for every r > 5 there exists a constant c > 0 such that for every a ∈ A P ∫ )−r ( )−r +5 k( Φ(an)− 4 log Φ(an) dn ≤ c 1 + ∥ log(a)∥ . aρP NP

(ii) If F = C, then for every r > 5 there exists a constant c > 0 such that for every a ∈ A Q and b ∈ A P ∩ M Q ∫ )−r k( ρP a Φ(abn)− 4 log Φ(abn) dn ≤ c∥ log a∥3 . NP

It follows from these estimates that for every φ ∈ C(G/H ) and g ∈ G the integral ∫ φ(gn) dn R P φ(g) := NP

is absolutely convergent and that the map R P thus obtained is a continuous linear map from C(G/H ) to C(G/(L P ∩ H )N P ). Since R P is equivariant and the left regular representation of G on C(G/H ) is smooth, R P in fact defines a continuous linear map from C(G/H ) to C ∞ (G/(L P ∩ H )N P ). By the estimates (i) and (ii) H Q defines a continuous linear map C(G/H ) → C(L P /L P ∩ H )′ ∞ and thus H P is in fact a continuous map to Ctemp (L P /L P ∩ H ). Finally, if F = R, then it follows as in the proof for maximal σ -parabolic subgroups in Section 2.3 that H P defines a continuous linear map C(G/H ) → C(L Q /L Q ∩ H ). The remaining assertion in case F = C is trivial as M P is compact. This ends the proof of Theorem 2.1. 3. Kernels Throughout this section we assume that F = R, C. 3.1. The Plancherel decomposition and kernels of Radon transforms For a general reductive symmetric space a description of the discrete series representations has been given by the first author [6] and Matsuki and Oshima [10]. In our setting, it follows from the rank condition in [10] that the space G/H does not admit discrete series. For a maximal σ parabolic subgroup Q, with Langlands decomposition Q = M Q A Q N Q such that M Q and A Q are σ -stable, the symmetric space M Q /(M Q ∩ H ) is non-compact and of rank 1 and hence admits discrete series representations if and only if M Q ∩ H is not compact. Delorme [5] and independently Van den Ban and Schlichtkrull [17,18] have given a precise description of the Plancherel decomposition of a general reductive symmetric space. It follows from these descriptions that in our setting the space L 2 (G/H ) decomposes as L 2 (G/H ) = L 2Pσ (G/H ) ⊕ L 2Qσ (G/H ),

(3.1)

where L 2S (G/H ) for S = Pσ , Qσ is a G-invariant closed subspace that is unitarily equivalent to a direct integral of representations that are induced from a parabolic subgroup contained in S. To be more precise, if S ∈ S and = M S A S N S is a Langlands decomposition of S such that A S is σ -stable, then L 2S (G/H ) is unitarily equivalent to a direct integral of representations ˆS is so that for some v ∈ N K (a) IndGS (ξ ⊗ λ ⊗ 1), where λ ∈ ia∗S and ξ ∈ M ( ) ( ) HomG ξ, L 2 M S /(M S ∩ v H v −1 ) ̸= {0}, i.e., ξ is equivalent to a discrete series representation for the space M S /(M S ∩ v H v −1 ). Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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Intersecting both sides of (3.1) with C(G/H ) yields a decomposition C(G/H ) = CPσ (G/H ) ⊕ CQσ (G/H ), where CPσ (G/H ) = C(G/H ) ∩ L 2Pσ (G/H ) and CQσ (G/H ) = C(G/H ) ∩ L 2Qσ (G/H ). For a σ -parabolic subgroup P, we denote the kernel of R P in C(G/H ) by ker(R P ). The aim of this section is to prove the following theorem. Theorem 3.1. ⋂ (i) ⋂ P∈Pσ ker(R P ) = CQσ (G/H ). (ii) Q∈Qσ ker(R Q ) = {0}. Remark 3.2. Let P be a σ -parabolic subgroup. If h ∈ H and P ′ = h Ph −1 , then ker(R P ) = ker(R P ′ ). Therefore, ⋂

⋂

ker(R P ) =

ker(R Pi ),

i∈{1,2,3}

P∈Pσ

⋂ Q∈Qσ

ker(R Q ) =

⋂

ker(R Q i ).

i∈{1,2,3,4}

We will prove the theorem by defining τ -spherical Harish-Chandra transforms, relating them to τ -spherical Fourier transforms as defined by Delorme in Section 3 of [5], and then using the Plancherel theorem to conclude the assertions. This program is carried out in the remainder of Section 3. 3.2. The τ -spherical Harish-Chandra transform Let (τ, Vτ ) be a finite dimensional representation of K . We write C ∞ (G/H : τ ) for the space of smooth Vτ -valued functions φ on G/H that satisfy the transformation property φ(kx) = τ (k)φ(x)

(k ∈ K , x ∈ G/H ).

We further write : τ ) and C(G/H : τ ) for the subspaces of C ∞ (G/H : τ ) consisting of compactly supported functions and Schwartz functions, respectively. Let W be the Weyl group of the root system of a in g. Then Cc∞ (G/H

W = N K (a)/Z K (a). For a subgroup S of G, we define W S to be the subgroup of W consisting of elements that can be realized in N K ∩S (a). Let P be a σ -parabolic subgroup containing A and let P = M P A P N P be a Langlands decomposition such that A P ⊆ A. We write W P for a choice of a set of representatives in N K (a) for the double quotient W P \ W/W H . Note that for the symmetric spaces considered here W P has at most 3 elements. We denote by τ M P the restriction of τ to M P and define ⨁ ( ) C P (τ ) := C M P /M P ∩ v H v −1 , τ M P , v∈W P

( ) If ψ ∈ C P (τ ), we write ψv for the component of ψ in the space C M P /M P ∩ v H v −1 , τ M P . For v ∈ W we define the parabolic subgroup P v by P v := v −1 Pv. Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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It follows from Theorem 2.1 that for every φ ∈ C(G/H, τ ), a ∈ A P and m ∈ M P the integral ∫ v ρP H P,τ φ(a)(m) := a φ(mavn) dn N Pv

is absolutely convergent. Moreover, for every a ∈ A P the function HvP,τ φ(a) belongs to C(M P /M P ∩ v H v −1 , τ M P ), and from 3a and 3b in Lemma 2.4 it follows that the map from A P to C(M P /M P ∩ v H v −1 , τ M P ) thus obtained is continuous and tempered in the sense that for every continuous seminorm µ on C(M P /M P ∩ v H v −1 , τ M P ) there exists an r > 0 so that ( ) sup (1 + ∥a∥)−r µ HvP,τ φ(a) < ∞. a∈A P

Definition 3.3. For a function φ ∈ C(G/H, τ ) we define its τ -spherical Harish-Chandra transform H P,τ φ to be the function A P → C P (τ ) given by ∫ ( ) ( ) φ(mavn) dn v ∈ WP , m ∈ MP , a ∈ A P . H P,τ φ(a) (m) := a ρ P v

NPv

3.3. The τ -spherical Fourier transform We continue with the assumptions and notation from the previous section. Let A2,P (τ ) be the subspace of C P (τ ) consisting of all elements ψ such that for every v ∈ W P the function ψv is finite under the action of the algebra D(M P /M P ∩v H v −1 ) of M P -invariant differential operators on M P /M P ∩ v H v −1 . For each v ∈ W P the space A(M P /M P ∩ v H v −1 , τ M P ) := {ψv : ψ ∈ A2,P (τ )} is finite dimensional. See [4, Proposition 1]. Equipped with the restriction of the inner product of L 2 (M P /M P ∩ v H v −1 , Vτ ) this space is therefore a Hilbert space. Since A2,P (τ ) is the direct sum of the spaces A(M P /M P ∩ v −1 H v, τ M P ), it is the finite direct sum of finite dimensional Hilbert spaces, and thus it is itself a finite dimensional Hilbert space. For ψ ∈ A2,P (τ ) and λ ∈ a∗P,C let ψλ : G/H → Vτ be the function given by ⎧ ⋃ ) ( ⎪ Pv H x ̸∈ ⎨0, v∈W P ψλ (x) := ⎪ ⎩a −λ+ρ P ψ (m), (x ∈ N amv H, a ∈ A , m ∈ M , v ∈ W ). v P P P P If Re (λ − ρ P ) is strictly P-dominant we define the (unnormalized) Eisenstein integral E(P, ψ, λ) : G/H → Vτ by ∫ E(P, ψ, λ)(x) = τ (k −1 )ψλ (kx) dk (x ∈ G/H ) K

and for other λ ∈ a∗P,C we define E(P, ψ, λ) by meromorphic continuation (see [3, Section 3]). The Eisenstein integral can be normalized by setting E 0 (P, ψ, λ) := E(P, C P|P (1, λ)−1 ψ, λ) ( ) as an identity of meromorphic functions in λ. Here C P|P (1, λ) ∈ End A2,P (τ ) is the c-function determined by the asymptotic expansion in [5, (3.3)] for E(P, ψ, λ) and is invertible for generic ∏ λ. The function λ ↦→ C P|P (1, λ) is meromorphic. In fact there exists a finite product b = nj=1 (⟨α j , · ⟩ − c j ) of factors ⟨α j , · ⟩ − c j , where α j ∈ Σ does not vanish on a P and Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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c j ∈ C, with the property that λ ↦→ b(λ)C P|P (1, λ) is holomorphic on an open neighborhood of ia∗P . See [3, Th´eor`eme 1]. 0 For φ ∈ Cc∞ (G/H, τ ) let F P,τ φ(λ) be the element of A2,P (τ ) determined by ∫ ⟨ ⟩ 0 ⟨F P,τ φ(λ), ψ⟩ = φ(x), E 0 (P, ψ, λ) τ d x. G/H

defines a continuous map C(G/H : τ ) → S (ia∗P ) ⊗ A2,P (τ ). The map By [3, Th´eor`eme 4] 0 F P,τ thus obtained is called the (normalized) τ -spherical Fourier transform. Let F A P be the euclidean Fourier transform on A P , i.e., the transform F A P : S (A P ) → S (ia∗P ) given by ∫ F A P f (λ) = f (a)a −λ da ( f ∈ S (A P ), λ ∈ ia∗P ). 0 F P,τ

AP

The continuous extension of F A P to a map S ′ (A P ) → S ′ (ia∗P ) we also denote by F A P . Lemma 3.4. Let φ ∈ C(G/H, τ ). Then ( ) 0 ⟨F P,τ φ(λ), ψ⟩ = F A P ⟨H P,τ φ( · ), C P|P (1, λ)−1 ψ⟩ (λ)

(

) λ ∈ ia∗P , ψ ∈ A2,P (τ ) .

as an identity of tempered distributions on ia∗P . Proof. Let ∑

Γ P :=

R≥0 Hα ,

α∈Σ (g,a) gα ⊆n P

where Hα ∈ a∗ is the element so that α(Y ) = ⟨Y, Hα ⟩

(Y ∈ a).

( ) Let B (⊆ a be so that supp(φ) ⊆ K exp(B)H . It follows from [8, Corollary 4.2] that supp H P,τ φ ) ⊆ exp (B + Γ P ) ∩ a P . Since H P,τ φ is a tempered C(M P /M P ∩ v H v −1 , τ M P )-valued function, it follows that for every t ∈ (0, 1] the function a ↦→ a −tρ P H P,τ φ(a) belongs to S (A P ) and the map [0, 1] → S ′ (A P );

t ↦→ ( · )−tρ P H P φ

is continuous. The proof is analogous to the proof of [8, Lemma 5.7]. For every ψ ∈ A2,P (τ ), λ ∈ ia∗P and t > 0 ( ) F A P ⟨H P,τ φ( · ), ψ⟩ (tρ P + λ) ∫ ∫ ∑ ∫ = a −λ+(1−t)ρ P ⟨φ(manv H ), ψv (m)⟩τ dm dn da AP

=

N P v∈W P

∑ ∫ v∈W P

∫

∫ M P /M P ∩v H v −1

M P /M P ∩v H v −1

∫ AP

⟨φ(manv H ), ψtρ P +λ (manv H )⟩τ dm dn da NP

⟨φ(x), ψtρ P +λ (x)⟩τ d x.

= G/H

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Since the measure on G/H is invariant, the last integral is equal to ∫ ∫ ⟨φ(kx), ψtρ P +λ (kx)⟩τ d x dk K G/H ∫ ∫ = ⟨φ(x), τ (k −1 )ψtρ P +λ (kx)⟩τ dk d x G/H K ∫ = ⟨φ(x), E(P, ψ, tρ P + λ)⟩τ d x. G/H

After replacing ψ by C P|P (1, tρ P + λ)−1 ψ, we have thus obtained the identity ( ) 0 ⟨F P,τ φ(tρ P + λ), ψ⟩ = F A P ⟨H P,τ φ( · ), C P|P (1, tρ P + λ)−1 ψ⟩ (tρ P + λ). The assertion in the lemma now follows by taking the limit for t ↓ 0 on both sides of the equation. □ 3.4. Proof of Theorem 3.1 ˆ, we denote by Vϑ For a K -invariant subspace V of L 2 (G/H ) and a finite subset ϑ of K the subspace of V of all K -finite vectors with isotypes contained in ϑ. By continuity and ˆ equivariance of the Radon transforms, it suffices to prove that for all finite subsets ϑ of K ⋂ ⋂ ker(R P )ϑ = CQσ (G/H )ϑ , ker(R Q )ϑ = {0}. (3.2) P∈Pσ

Q∈Qσ

Let C(K )ϑ be the space of K -finite continuous functions on K , whose right K -types belong to ϑ and let τ denote the right regular representation of K on Vτ := C(K )ϑ . Then the canonical map ς : C(G/H )ϑ → C(G/H : τ ) given by ς φ(x)(k) = φ(kx)

(

φ ∈ C(G/H )ϑ , k ∈ K , x ∈ G/H

)

is a linear isomorphism. Let φ ∈ C(G/H )ϑ and let P be a σ -parabolic ( subgroup.)Then R P v φ = 0 for every v ∈ W P if and only if H P,τ (ς φ) = 0. The space ς CPσ (G/H )ϑ is ( equal to the) space C(G/H, τ )(P) defined in Th´eor`eme 2 in [5], where P ∈ Pσ . Likewise, ς CQσ (G/H )ϑ equals C(G/H, τ )(Q) with Q ∈ Qσ . (For the symmetric spaces considered here there are three association classes of σ -parabolic subgroups: Pσ , Qσ and {G}.) In addition we have C(G/H, τ )(Q) ∩ C(G/H, τ )(P) = {0}.

(3.3)

It follows from Lemma 3.4 and the fact that the c-function C P|P (1, λ) is invertible for generic λ that 0 ker(H P,τ ) ⊆ ker(F P,τ ).

(3.4)

For minimal σ -parabolic subgroups P the spaces A2,P (τ ) and C P (τ ) coincide and therefore the inclusion (3.4) is an equality for these parabolic subgroups P. 0 Note that FG,τ = {0} since G/H does not admit discrete series representation. It now follows from [5, Th´eor`eme 2] that if P ∈ Pσ and Q ∈ Qσ , then ker(H P,τ ) = C(G/H, τ )(Q) ,

ker(H Q,τ ) ⊆ C(G/H, τ )(P) .

Moreover, the identity (2.6) implies that ker(H Q,τ ) ⊆ ker(H P,τ ). The proof for (3.2) and thus for Theorem 3.1 is now concluded by observing that (3.3) implies ker(H Q,τ ) = {0}. Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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4. Divergence of cuspidal integrals for SL(3, H)/Sp(1, 2) In this section, let F be equal to H. For y, z ∈ H, let n y,z be given by (1.7) and let Φ be given by (1.4). Lemma 4.1. Let ϵ < 16 . Then the integral ∫ ∫ Φ(n y,z )−1−ϵ dy dz H

H

is divergent. Proof. It follows from Lemma 1.11 that Φ(n y,z ) is equal to ( ) (1 − |z|2 )2 + (1 + |y|2 )2 + 1 + 2|y|2 + 2|z|2 + 2|y|2 |z|2 ( ) 2 + (1 + |y|2 − |z|2 )2 + 2|y|2 + 2|z|2 . There exists a constant c > 0 such that for every y, z ∈ H satisfying 1 ≤ |y| ≤ |z| ≤ |y| + 1 we have (1 − |z|2 )2 + (1 + |y|2 )2 + 1 + 2|y|2 + 2|z|2 + 2|y|2 |z|2 ≤ c|y|4 and 2 + (1 + |y|2 − |z|2 )2 + 2|y|2 + 2|z|2 ≤ c|y|2 . Then there exists a constant C > 0 such that ∫ ∫ ∫ ∫ −1−ϵ Φ(n y,z ) dy dz ≥ H

y∈H |y|≥1

H

z∈H |y|≤|z|≤|y|+1

Φ(n y,z )

−1−ϵ

∞

∫

r 6r −6−6ϵ dr.

dy dz ≥ C 1

The latter integral is divergent for ϵ < 61 . □ Proposition 4.2. There exists a positive function φ ∈ C(G/H ) such that for every Q ∈ Qσ that is H -conjugate to Q 1 or Q 4 and for every g ∈ G the integral ∫ φ(gn) dn NQ

is divergent. Proof. Let Ξ be Harish-Chandra’s bi-K -invariant spherical function φ0 on G. Define Θ to be the function G/H → R+ given by √ Θ(x) = Ξ (gσ (g)−1 ) (x = g H ∈ G/H ). Let 0 < ϵ < 61 and let φ = Θ −1−ϵ . It follows from Propositions 17.2 and 17.3 in [13] that φ ∈ C(G/H ) and that for every g ∈ G there exists a c > 0 such that φ(gx) ≥ cΦ(x)−1−ϵ

(x ∈ G/H ).

For Q = Q 1 the proposition now follows from Lemma 4.1; for Q = Q 4 it then follows from Proposition 1.7. If h ∈ H and Q = h Q i h −1 with i = 1, 4, then the integral ∫ ∫ ∫ −1 φ(gn) dn = φ(ghnh ) dn = φ(ghn) dn NQ

N Qi

N Qi

is divergent. This proves the claim for parabolic subgroups that are H -conjugate to Q 1 or Q 4 . □ Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.

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Corollary 4.3. Let φ ∈ C(G/H ) be as in Proposition 4.2. For every P ∈ Pσ and every g ∈ G the integral ∫ φ(gn) dn NP

is divergent. Proof. Note that n Q 1 ⊂ p1 . Let g ∈ G. By Tonelli’s theorem ∫ ∫ ∫ φ(gn) dn = φ(gnn 1 ) dn 1 dn. N P1

N P1 /N Q 1

N Q1

It follows from Proposition 4.2 and Fubini’s theorem that the right-hand side is divergent. Similarly we have n Q 4 ⊂ p2 ∩ p3 and thus we see that the integrals for P = P2 and P = P3 are divergent as well. The assertion now follows from Proposition 1.3. □ References [1] N.B. Andersen, M. Flensted-Jensen, Cuspidal discrete series for projective hyperbolic spaces, in: Geometric Analysis and Integral Geometry, in: Contemp. Math., vol. 598, Amer. Math. Soc., Providence, RI, 2013, pp. 59–75. [2] N.B. Andersen, M. Flensted-Jensen, H. Schlichtkrull, Cuspidal discrete series for semisimple symmetric spaces, J. Funct. Anal. 263 (8) (2012) 2384–2408. [3] J. Carmona, P. Delorme, Transformation de Fourier sur l’espace de Schwartz d’un espace symétrique réductif, Invent. Math. 134 (1) (1998) 59–99. [4] P. Delorme, Troncature pour les espaces symétriques réductifs, Acta Math. 179 (1) (1997) 41–77. [5] P. Delorme, Formule de Plancherel pour les espaces symétriques réductifs, Ann. of Math. (2) 147 (2) (1998) 417–452. [6] M. Flensted-Jensen, Discrete series for semisimple symmetric spaces, Ann. of Math. (2) 111 (2) (1980) 253–311. [7] A.W. Knapp, Lie Groups Beyond an Introduction, second ed., in: Progress in Mathematics, vol. 140, Birkhäuser, 2002. [8] J.J. Kuit, Radon transformation on reductive symmetric spaces: support theorems, Adv. Math. 240 (2013) 427–483. [9] T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (2) (1979) 331–357. [10] T. Oshima, T. Matsuki, A description of the discrete series for semisimple symmetric spaces, Adv. Stud. Pure Math. 4 (1984) 331–390. [11] T. Oshima, J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1) (1980) 1–81. [12] W. Rossmann, The structure of semisimple symmetric spaces, Canad. J. Math. 31 (1) (1979) 157–180. [13] E.P. van den Ban, The principal series for a reductive symmetric space. II. Eisenstein integrals, J. Funct. Anal. 109 (2) (1992) 331–441. [14] E.P. van den Ban, J.J. Kuit, Cusp forms for reductive symmetric spaces of split rank one, Represent. Theory 21 (2017) 467–533. [15] E.P. van den Ban, J.J. Kuit, New normalizations for eisenstein integrals, J. Funct. Anal. 272 (7) (2017) 2795–2864. [16] E.P. van den Ban, J.J. Kuit, H. Schlichtkrull, The notion of cusp forms for a class of reductive symmetric spaces of split rank one, Kyoto J. Math (2018) arxiv:1406.1634. (in press). [17] E.P. van den Ban, H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space I Spherical Functions, Invent. Math. 161 (3) (2005) 453–566. [18] E.P. van den Ban, H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space II. Representation Theory, Invent. Math. 161 (3) (2005) 567–628.

Please cite this article in press as: M. Flensted-Jensen, J.J. Kuit, Cuspidal integrals for SL(3)/K ϵ , Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.05.005.